In the presence of chance moves in a semantical game, the existence of pure optimal strategies does not guarantee the existence of winning ones. This fact provides a basis for constructing supervaluational semantical games with a chance move. Additional possibilities of using chance moves in game-theoretical semantics are also discussed.

We associate the semantic game with chance moves conceived by Blinov with Blamey’s partial logic. We give some equivalent alternatives to the semantic game, some of which are with a third player, borrowing the idea of introducing the pseudo-player called Nature in game theory. We observe that IF propositional logic proposed by Sandu and Pietarinen can be equivalently translated to partial logic, which implies that imperfect information may not be necessary for IF propositional logic. We also (...) indicate that some independent quantifiers can be regarded as dependent quantifiers of indeterminate sequence, using the interjunction connective in partial logic. We conclude our paper by indicating some further research in a more general setting. (shrink)

We consider two versions of truth as grounded in verification procedures: Dummett's notion of proof as an effective way to establish the truth of a statement and Hintikka's GTS notion of truth as given by the existence of a winning strategy for the game associated with a statement. Hintikka has argued that the two notions should be effective and that one should thus restrict one's attention to recursive winning strategies. In the context of arithmetic, we show that the two (...) notions do not coincide: on the one hand, proofs in PA do not yield recursive winning strategies for the associated game; on the other hand, there is no sound and effective proof procedure that captures recursive GTS truths. We then consider a generalized version of Game Theoretical Semantics by introducing games with backward moves. In this setting, a connection is made between proofs and recursive winning strategies. We then apply this distinction between two kinds of verificationist procedures to a recent debate about how we recognize the truth of Gödelian sentences. (shrink)

We present an axiomatic approach for a class of finite, extensive form games of perfect information that makes use of notions like “rationality at a node” and “knowledge at a node.” We distinguish between the game theorist's and the players' own “theory of the game.” The latter is a theory that is sufficient for each player to infer a certain sequence of moves, whereas the former is intended as a justification of such a sequence of moves. While (...) in general the game theorist's theory of the game is not and need not be axiomatized, the players' theory must be an axiomatic one, since we model players as analogous to automatic theorem provers that play the game by inferring (or computing) a sequence of moves. We provide the players with an axiomatic theory sufficient to infer a solution for the game (in our case, the backwards induction equilibrium), and prove its consistency. We then inquire what happens when the theory of the game is augmented with information that a move outside the inferred solution has occurred. We show that a theory that is sufficient for the players to infer a solution and still remains consistent in the face of deviations must be modular. By this we mean that players have distributed knowledge of it. Finally, we show that whenever the theory of the game is group-knowledge (or common knowledge) among the players (i.e., it is the same at each node), a deviation from the solution gives rise to inconsistencies and therefore forces a revision of the theory at later nodes. On the contrary, whenever a theory of the game is modular, a deviation from equilibrium play does not induce a revision of the theory. (shrink)

Understanding human behaviour involves "why"'s as well as "how"'s. Rational people have good reasons for acting, but it can be hard to find out what these were and how they worked. In this Note, we discuss a few ways in which actions, preferences, and expectations are intermingled. This mixture is especially clear with the well-known solution procedure for extensive games called 'Backward Induction'. In particular, we discuss three scenarios for analyzing behaviour in a game. One can rationalize (...) given moves as revealing agents' preferences, one can also rationalize them as revealing agents' beliefs about others, but one can also change a predicted pattern of behaviour by making promises. All three scenarios transform given games to new ones, and we prove some results about their scope. A more general view of relevant game transformations would involve dynamic and epistemic game logics. Finally, our analysis describes and disentangles matters: but it will not tell you what to do! (shrink)

In order to refute the widely held belief that the game known as ‘Newcomb's paradox’ is physically nonsensical and impossible to imagine (e.g. because it involves backward causation), I tell a story in which the game is realized in a classical, deterministic universe in a physically plausible way. The predictor is a collection of beings which are by many orders of magnitude smaller than the player and which can, with their exquisite measurement techniques, observe the particles in the (...) player's body so accurately that they can predict his choice (in much the same way as we can predict the motion of celestial bodies). I argue that the player, by choosing whether to take only one box or both boxes, influences whether or not, in the past, the predictor put a million pounds into the second box. Yet, I establish that no causal paradox can arise in this set-up. (shrink)

According to the standard objection to backward induction in games, its application depends on highly questionable assumptions about the players' expectations as regards future counterfactual game developments. It seems that, in order to make predictions needed for backward reasoning, the players must expect each player to act rationally at each node that in principle could be reached in the game, and also to expect that this confidence in the future rationality of the players would be kept by (...) each player come what may: even at the game-nodes that could only be reached by irrational play. Both expectations seem to be rather unreasonable: a player's initial disposition to rational behaviour may be weakened by a long stretch of irrational play on his part and, even more importantly, his initial confidence in the other players' future rationality may be undermined by an irrational play on their part. For different formulations of this objection see Binmore, Reny and, Bicchieri, Pettit and Sugden. and Aumann and.). (shrink)

We observe that in certain two-player repeated games of incomplete information, where information may be incomplete on both sides, it is possible for an informed player to signal his status as an informed player to the other without revealing any information about the choice of chance. The key to obtaining such a class of games is to relax the assumption that the players’ moves are observable. We show that in such cases players can achieve a kind of (...) signaling that is “zero-knowledge”, in the sense that the other player becomes convinced that her opponent is informed without ever learning the choice of chance. Moreover, such “zero-knowledge signaling” has all of the statistical properties associated with zero-knowledge proofs in intereactive protocols. In particular, under the general assumption that moves are unobservable, such signaling leads to a class of equilibria in repeated games that are separatingin regard to the status of player 1–informed or uninformed–but only for player 2; any other player in a network, being unable to observe the moves of player 2, remains uncertain as to the status of player 1. (shrink)

The standard backward-induction reasoning in a game like the centipede assumes that the players maintain a common belief in rationality throughout the game. But that is a dubious assumption. Suppose the first player X didn't terminate the game in the first round; what would the second player Y think then? Since the backwards-induction argument says X should terminate the game, and it is supposed to be a sound argument, Y might be entitled to doubt X's rationality. Alternatively, Y might (...) doubt that X believes Y is rational, or that X believes Y believes X is rational, or Y might have some higher-order doubt. X’s deviant first move might cause a breakdown in common belief in rationality, therefore. Once that goes, the entire argument fails. The argument also assumes that the players act rationally at each stage of the game, even if this stage could not be reached by rational play. But it is also dubious to assume that past irrationality never exerts a corrupting influence on present play. However, the backwards-induction argument can be reconstructed for the centipede game on a more secure basis.1 It may be implausible to assume a common belief in rationality throughout the game, however the game might go, but the argument requires less than this. The standard idealisations in game theory certainly allow us to assume a common belief in rationality at the beginning of the game. They also allow us to assume this common belief persists so long as no one makes an irrational move. That is enough for the argument to go through. (shrink)

Backward induction has been the standard method of solving finite extensive-form games with perfect information, notwithstanding the fact that this procedure leads to counter-intuitive results in various games. However, beginning in the late eighties, the method of backward induction became an object of criticism. It is claimed that the assumptions needed for its defence are quite implausible, if not incoherent. It is therefore natural to ask for the justification of backward induction: Can one show (...) that rational players who know the structure of the game, have trust in each other's practical rationality and reason correctly, will act in accordance in backward induction? Several researchers have tried a justification of this kind, but the argument presented in Robert Aumann's paper from 1995 is perhaps the most well-known and influential attempt to provide such a justification. Clausing provides a sustained discussion of the justification problem for backward induction. It is an excellent work and the criticism I will present below does not detract from this evaluation: the issues discussed by the author are complex and it is difficult to get everything right. Furthermore, I hope that the criticism to be presented may be instructive; Even though it has not been Clausing's intention, his logical reconstruction of Aumann's defence of backward induction allows us to see very clearly what is wrong with that argument. It also provides us with * This paper was presented at workshops in Lund and in Uppsala, in the Fall of 1999. I am indebted to the participants for their useful comments. The work on the paper was supported by a generous research grant from The Bank of Sweden Tercentenary Foundation. (shrink)

We do not believe that logic is the sole answer to deep and intriguing questions about human behaviour, but we think that it might be a useful tool in simulating and understanding it to a certain degree and in specifically restricted areas of application. We do not aim to resolve the question of what rational behaviour in games with mistaken and changing beliefs is. Rather, we develop a formal and abstract framework that allows us to reason about behaviour (...) in games with mistaken and changing beliefs leaving aside normative questions concerning whether the agents are behaving “rationally”; we focus on what agents do in a game. In this paper, we are not concerned with the reasoning process of the (ideal) economic agent; rather, our intended application is artificial agents, e.g., autonomous agents interacting with a human user or with each other as part of a computer game or in a virtual world. We give a story of mistaken beliefs that is a typical example of the situation in which we should want our formal setting to be applied. Then we give the definitions for our formal system and how to use this setting to get a backward induction solution. We then apply our semantics to the story related earlier and give an analysis of it. Our final section contains a discussion of related work and future projects. We discuss the advantages of our approach over existing approaches and indicate how it can be connected to the existing literature. (shrink)

Two players bargain to select a utility allocation in some set X⊂R+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\subset {\mathbb {R}}_+^2$$\end{document}. Bargaining takes place in infinite discrete time, where each period t is divided into two sub-periods. In the first sub-period, the players play a simultaneous-move game to determine that period’s proposer, and bargaining takes place in the second sub-period. Rejection triggers a one-period delay and move to t+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t+1$$\end{document}. For (...) every x∈X∩R++2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in X\cap {\mathbb {R}}^2_{++}$$\end{document}, there exists a cutoff δ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta <1$$\end{document}, such that if at least one player has a discount factor above δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}, then for every y∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\in X$$\end{document} that satisfies y≥x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\ge x$$\end{document} there exists a subgame perfect equilibrium with immediate agreement on y. The equilibrium is supported by “dictatorial threats.” These threats can be dispensed with if X is the unit simplex and the target-vector is Pareto efficient. The results can be modified in a way that allows for arbitrarily long delays in equilibrium. (shrink)

As the field of clinical bioethics has moved from its pioneers, who turned their attention to ethics problems in clinical medicine and clinical and animal research, to today’s ubiquity of university degrees and fellowships in bioethics, there has been a steady drumbeat to professionalize the field. The problem has been that the necessary next steps—to specify the skills, knowledge, and personal and professional attributes of a clinical bioethicist, and to have a method to train and evaluate mastery of these standards—are (...) lacking.Ordinarily, the path to professionalism in medicine starts with the intellectual pioneers. Then come those who develop early experience in clinical settings. Then comes the specification of the skills, knowledge, and personal and professional attributes needed to perform the activities of the new specialty or subspecialty. And only then, after a method to train and evaluate levels of mastery from novice to advanced practitioner has been developed, comes credible credentialing and certification/licensing.Unfortunately, the field of clinical bioethics has skipped these steps. Rather, a credential, that is, the Healthcare Ethics Consultant (HEC) certification, was created by a small group within the bioethics professional association, the American Society of Bioethics and Humanities (ASBH), without community agreement or necessary input. Further, the testing of processes to train HECs and to evaluate their levels of mastery of competencies was prematurely forwarded as sufficient evidence of competence in clinical ethics. That is, the credential, offered by the ASBH for a fee, upon passing an exam based on how many hours one has been involved in clinical consultation, about which there is no field agreement on how such consultations ought to be conducted and for which only controversial standards have been set, is being touted as evidence of competence in clinical bioethics. In their article, “Competencies and Milestones for Bioethics Trainees,” Sawyer and colleagues identify the central weakness of these claims to professionalization and provide the field the first substantive assessment tool and method to train and evaluate competencies. The tool these authors present is the real next step forward for true professionalization of the field of clinical bioethics. (shrink)

Despite the challenges associated with virtually mediated communication, remote collaboration is a defining characteristic of online multiplayer gaming communities. Inspired by the teamwork exhibited by players in first-person shooter games, this study investigated the verbal and behavioral coordination of four-player teams playing a cooperative online video game. The game, Desert Herding, involved teams consisting of three ground players and one drone operator tasked to locate, corral, and contain evasive robot agents scattered across a large desert environment. Ground players (...) could move throughout the environment, while the drone operator’s role was akin to that of a “spectator” with a bird’s-eye view, with access to veridical information of the locations of teammates and the to-be-corralled agents. Categorical recurrence quantification analysis was used to measure the communication dynamics of teams as they completed the task. Demands on coordination were manipulated by varying the ground players’ ability to observe the environment with the use of game “fog.” Results show that catRQA was sensitive to changes to task visibility, with reductions in task visibility reorganizing how participants conversed during the game to maintain team situation awareness. The results are discussed in the context of future work that can address how team coordination can be augmented with the inclusion of artificial agents, as synthetic teammates. (shrink)

In the target article Nijhawan speculates that visual perceptual mechanisms compensate for neural delays so that moving objects may be perceived closer to their physical locations. However, the vast majority of published psychophysical data are inconsistent with this speculation.

Peirce believed that his pragmaticism can be conclusively proven. Beginning in 1903, he drafted several attempts, ending by 1908 with a semeiotic proof. Around 1905, he exposes the proof using the theory of Existential Graphs . This paper modernizes the semantics Peirce proposed for EGs in terms of game-theoretic semantics . Peirce's 1905 proof is then reconstructed in three parts, by relating pragmaticism to the GTS conception of meaning, showing that Peirce's proof is an argument for a relational structure (...) of the meaning of intellectual signs that our interpretative and strategic practices give rise to, and bringing out the key links between EGs and pragmaticism. (shrink)

Through a phenomenological study of horse-human relations, this article explores the ways in which, as embodied beings, we live relationally, rather than as separate human identities. Conceptually this challenges oppositional logic and humanist assumptions, but where poststructuralist treatments of these issues tend to remain abstract, this article is concerned with an embodied demonstration of the ways in which we experience a relational or in-between logic in our everyday lives.

We now move to the demonstration of the left-to-right direction of the equivalence result. Let us assume that there is a winning $$\mathbf {P}$$ P -strategy in the dialogical game for $$\varphi $$ φ.

If semantic consequence is analyzed with extensive games, logical reasoning can be accounted for by looking at how players solve entailment games. However, earlier approaches to game semantics cannot achieve this reduction, by want of explicitly dened preferences for players. Moreover, although entailment games can naturally translate the idea of argumentation about a common ground, a cognitive interpretation is undermined by the complexity of strategic reasoning. We thus describe a class of semantic extensive entailment game (...) class='Hi'>with algorithmic players, who have preferences for parsimonious spending of computational resources and thus compute partial strategies under qualitative uncertainty about future histories. We prove the existence of local preferences for moves and of strategic fixpoints that allow to map game-trees to tableaux proofs, and exhibit a strategy prole that solves the fixpoint selection problem, and can be mapped to systematic constructions of semantic trees, yielding a completeness result by translation. We then discuss the correspondence between proof heuristics and strategies in our games, the relations of our games to gts, and possible extensions to other entailment relations. We conclude that the main interest of our result lies in the possibility to bridge argumentative and cognitive models of logical reasoning, rather than in new meta-theoretic results. All proofs are given in appendix. (shrink)

Theory-of-mind (ToM) involves modeling an individual’s mental states to plan one’s action and to anticipate others’ actions through recursive reasoning that may be myopic (with limited recursion) or predictive (with full recursion). ToM recursion was examined using a series of two-player, sequential-move matrix games with a maximum of three steps. Participants were assigned the role of Player I, controlling the initial and the last step, or of Player II, controlling the second step. Appropriate for the assigned (...) role, participants either anticipated or planned Player II’s strategy at the second step, and then determined Player I’s optimal strategy at the first step. Participants more readily used predictive reasoning as Player II (i.e., planning one’s own move) than as Player I (i.e., anticipating an opponent’s move), although they did not differ when translating reasoning outcome about the second step to optimal action in the first step. Perspective-taking influenced likelihood of predictive reasoning, but it did not affect the rate at which participants acquired it during the experimental block. We conclude that the depth of ToM recursion (related to perspective-taking mechanisms) and rational application of belief–desire to action (instrumental rationality) constitute separate cognitive processes in ToM reasoning. (shrink)

Those are the words of Pope John Paul II, speaking in March 2004 to an international congress held in Rome. The conference was on "Life-sustaining Treatments and Vegetative State: Scientific Advances and Ethical Dilemmas," and it was organized by the World Federation of Catholic Medical Associations and the Pontifical Academy for Life. The pope was able to cut through all the ethical dilemmas. Although he acknowledged that a patient in a persistent vegetative state, or PVS, "shows no evident sign of (...) selfawareness or of awareness of the environment, and seems unable to interact with others or to react to specific stimuli," he said that they should be kept alive indefinitely. Such patients, he insisted, "retain their human dignity in all its fullness" and "the loving gaze of God the Father continues to fall upon them." For this reason, he said, it is obligatory to continue to provide them with food and water, even if this can only be done through a tube. The pope added that to withdraw the tube, knowing that it will lead to the death of the patient, is "euthanasia by omission.". (shrink)

The pope supported his conclusion by arguing that some patients with PVS make at least a partial recovery, and, in the current state of medical science, we are still unable to predict with certainty which patients will recover and which will not. But here he seems to have been poorly advised. While it is true that in most PVS cases, we cannot definitively exclude the possibility of recovery, modern brain-imaging techniques do now enable us to know that in (...) some PVS cases, the entire cortex has been destroyed. Then, no recovery is possible, for the cortex cannot reconstitute itself. Hence the argument for preserving the lives of these patients cannot be based on medical uncertainty. (shrink)

In this essay, I address the question of whether the indisputable progress being made by the neurosciences poses a genuine threat to the language game of responsible agency. I begin by situating free will as an ineliminable component of our practices of attributing responsibility and holding one another accountable, illustrating this via a discussion of legal discourse regarding the attribution of responsibility for criminal acts. I then turn to the practical limits on agents' scientific self-objectivation, limits that turn out to (...) be mirrored philosophically in the conceptual problems that plague reductionist strategies. Having shown that free will is rooted in unavoidable performative presuppositions belonging to agents' participant perspective, I then take up the difficult issue of how to reconcile an epistemic dualism of participant and observer perspectives with the assumption of ontological monism. I critically review a range of proposed physicalist solutions, including non-reductionist and (standard) compatibilist approaches. An underlying problem with scientistic, physicalist approaches is the methodological fiction of an exclusive ?view from nowhere? which relies on the problematic move of disengaging the objectivating perspective of the scientific observer from the investigators' participant perspective of those engaged in scientific practice. Since there is no way of getting around the requisite complementarity of both the observer's encounter with the objective world and the participant's involvement in shared lifeworld practices, the remaining option is to take an epistemological turn. But even the recognition that science is ultimately constituted from within the lifeworld still leaves us with the question as to how the human mind can understand itself as the product of natural evolution. I conclude with some tentative suggestions as to how this difficult question might be addressed. (shrink)

Embodiment approaches to cognition and emotion have put forth the idea that the way we think and talk about affective events often recruits spatial information that stems, to some extent, from our bodily experiences. For example, metaphorical expressions such as “being someone’s right hand” or “leaving something bad behind” convey affectivity associated with the lateral and sagittal dimensions of space. Action tendencies associated with affect such as the directional fluency of hand movements (dominant right hand-side – positive; non-dominant (...) left hand-side – negative) and approach-avoidance behaviors (forward – positive; backwards – negative) might be mechanisms supporting such associations. Against this background, experimental research has investigated whether positive and negative words are freely allocated into space (e.g., close or far from one’s body) or resonate with congruent (vs. incongruent) predefined manual actions usually performed by joysticks or button presses (e.g., positive - right; negative - left, or vice versa). However, to date, it is unclear how the processing of affective concepts resonate with directional actions of the whole body that are performed freely within a context enabling lateral and sagittal movements simultaneously. Accordingly, 67 right-handed participants were to freely step on an 8-response pad (front, back, right, left, front-right, front-left, back-right, or back-left) after being presented in front of them valence-laden personal life-events submitted before the task (e.g., words or sentences such as “graduation” or “birth of a child”). The most revealing finding of this study indicates that approach-avoidance behaviors and space-valence associations across laterality are interwoven during whole body step actions: Positive events induced steps highly biased to front-right whereas negative events induced steps highly biased to back-left. (shrink)

In the context of propositional logics, we apply semantics modulo satisfiability—a restricted semantics which comprehends only valuations that satisfy some specific set of formulas—with the aim to efficiently solve some computational tasks. Three possible such applications are developed.We begin by studying the possibility of implicitly representing rational McNaughton functions in Łukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations in the (...) newly introduced system. An implementation of the algorithm, test results and ways to randomly generate rational McNaughton functions for testing are presented. Moreover, we propose an application of such representations to the formal verification of properties of neural networks by means of the reasoning framework of Łukasiewicz Infinitely-valued Logic.Then, we move to the investigation of the satisfiability of joint probabilistic assignments to formulas of Łukasiewicz Infinitely-valued Logic, which is known to be an NP-complete problem. We provide an exact decision algorithm derived from the combination of linear algebraic methods with semantics modulo satisfiability. Also, we provide an implementation for such algorithm for which the phenomenon of phase transition is empirically detected.Lastly, we study the game theory situation of observable games, which are games that are known to reach a Nash equilibrium, however, an external observer does not know what is the exact profile of actions that occur in a specific instance; thus, such observer assigns subjective probabilities to players actions. We study the decision problem of determining if a set of these probabilistic constraints is coherent by reducing it to the problems of satisfiability of probabilistic assignments to logical formulas both in Classical Propositional Logic and Łukasiewicz Infinitely-valued Logic depending on whether only pure equilibria or also mixed equilibria are allowed. Such reductions rely upon the properties of semantics modulo satisfiability. We provide complexity and algorithmic discussion for the coherence problem and, also, for the problem of computing maximal and minimal probabilistic constraints on actions that preserves coherence.prepared by Sandro Márcio da Silva Preto.E-mail: [email protected]:https://doi.org/10.11606/T.45.2021.tde-17062021-163257. (shrink)

Backward induction was one of the earliest methods developed for solving finite sequential games with perfect information. It proved to be especially useful in the context of Tom Schelling’s ideas of credible versus incredible threats. BI can be also extended to solve complex games that include an infinite number of actions or an infinite number of periods. However, some more complex empirical or experimental predictions remain dramatically at odds with theoretical predictions obtained by BI. The (...) primary example of such a troublesome game is Centipede. The problems appear in other long games with sufficiently complex structure. BI also shares the problems of subgame perfect equilibrium and fails to eliminate certain unreasonable Nash equilibria. (shrink)

Let Mn♯ denote the minimal active iterable extender model which has n Woodin cardinals and contains all reals, if it exists, in which case we denote by Mn the class-sized model obtained by iterating the topmost measure of Mn class-many times. We characterize the sets of reals which are Σ1-definable from R over Mn, under the assumption that projective games on reals are determined:1. for even n, Σ1Mn=⅁RΠn+11;2. for odd n, Σ1Mn=⅁RΣn+11.This generalizes a theorem of Martin and Steel for (...) L, that is, the case n=0. As consequences of the proof, we see that determinacy of all projective games with moves in R is equivalent to the statement that Mn♯ exists for all n∈N, and that determinacy of all projective games of length ω2 with moves in N is equivalent to the statement that Mn♯ exists and satisfies AD for all n∈N. (shrink)

A cornerstone of game theory is backward induction, whereby players reason backward from the end of a game in extensive form to the beginning in order to determine what choices are rational at each stage of play. Truels, or three-person duels, are used to illustrate how the outcome can depend on (1) the evenness/oddness of the number of rounds (the parity problem) and (2) uncertainty about the endpoint of the game (the uncertainty problem). Since there is no known (...) endpoint in the latter case, an extension of the idea of backward induction is used to determine the possible outcomes. The parity problem highlights the lack of robustness of backward induction, but it poses no conflict between foundational principles. On the other hand, two conflicting views of the future underlie the uncertainty problem, depending on whether the number of rounds is bounded (the players invariably shoot from the start) or unbounded (they may all cooperate and never shoot, despite the fact that the truel will end with certainty and therefore be effectively bounded). Some real-life examples, in which destructive behavior sometimes occurred and sometimes did not, are used to illustrate these differences, and some ethical implications of the analysis are discussed. (shrink)

Toward the end of Light in August, in the climactic scene in Chapter 1 where the authorities of justice pursue the elusive Joe Christmas through the streets of Jefferson, William Faulkner introduces a new character, Percy Grimm, a twenty-five-year-old captain in the State National Guard who has relentlessly acquired the rank of a special deputy for the search. As the town closes for the weekend, Grimm keeps vigil at a downtown store where other townsfolk have begun a poker game to (...) stay awake through the night as the search goes on. In their zeal to uphold law and justice, his men revel in their fantastic make-believe that they are doing the work of “a hidden and unsleeping and omnipotent eye watching the doings of men.” The poker game goes on through Saturday night until Christmas is spotted and given chase. As Grimm runs through the streets after the fleeing man, Faulkner presents to us the deputy's stream of thought from an omniscient point of view: “There was nothing vengeful about him either, no fury, no outrage. He was moving again almost before he had stopped, with that lean, swift, blind obedience to whatever Player moved him on the Board” (437). Through the rest of the chapter, Grimm, who has not participated in the poker game at all, thinks and acts as if he is engaged in a chess game, not as a player but as the stake or the pawn in a board game in which Jefferson is the board on which he and Christmas are being moved from place to place by two larger forces, he by the benevolent Providence and Christmas by the powers opposed to God. Finally, when Christmas is cornered and fatally shot in Reverend Hightower's kitchen, this fantasized game seems to end. (shrink)

An important approach to game theory is to examine the consequences of beliefs that agents may have about each other. This paper investigates respect for public preferences. Consider an agent A who believes that B strictly prefers an option a to an option b. Then A respects B’s preference if A assigns probability 1 to the choice of a given that B chooses a or b. Respect for public preferences requires that if it is common belief that B prefers a (...) to b, then it is common belief that all other agents respect that preference. Along the lines of Blume, Brandenburger and Dekel [3] and Asheim [1], I treat respect for public preferences as a constraint on lexicographic probability systems. The main result is that given respect for public preferences and perfect recall, players choose in accordance with Iterated Backward Inference. Iterated Backward Inference is a procedure that generalizes standard backward induction reasoning for games of both perfect and imperfect information. From Asheim’s characterization of proper rationalizability [1] it follows that properly rationalizable strategies are consistent with respect for public preferences; hence strategies eliminated by Iterated Backward Inference are not properly rationalizable. (shrink)

Abstract In this article we develop a relational understanding of sociality, that is, an account of social life that takes relation as primary. This stands in contrast to the common assumption that relations arise when subjects interact, an account that gives logical priority to separation. We will develop this relational understanding through a reading of the work of Martin Buber, a social philosopher primarily interested in dialogue, meeting, relationship, and the irreducibility and incomparability of reality. In particular, the article contrasts (...) Buber’s work with that of poststructuralist theorists who take as their starting point the deconstruction of the Hegelian logic of binary oppositions. Deconstruction understands difference as the excess that undoes the binary, but Buber, we argue, shows how difference derives from the primacy and ontological undefinability of relation. Relational logic does not exclude the logic of separations and oppositions: relation is the primal ground that makes separations possible. Content Type Journal Article Pages 1-13 DOI 10.1007/s11841-011-0278-9 Authors Andrew Metcalfe, School of Social Sciences and International Studies, University of New South Wales, Sydney, NSW 2052, Australia Ann Game, School of Social Sciences and International Studies, University of New South Wales, Sydney, NSW 2052, Australia Journal Sophia Online ISSN 1873-930X Print ISSN 0038-1527. (shrink)

Games, which philosophers commonly invoke as models for diverse phenomena, are plausibly understood in terms of rules and goals, but this gives rise to two puzzles. The first concerns the identity of a single game over time. Intuitively one and the same game can undergo a change in rules, as when the rules of chess were modified so that a pawn could be moved two squares forward on its first move. Yet if games are individuated in terms of (...) their constitutive rules and goals, this is incoherent—new rules mean a new game. The second concerns the individuation of games at a point in time. Intuitively, there can be different versions of a single game, where the versions differ in the details of their rules. I offer a solution to this problem that draws on an analogy with individuating languages. The resulting theory should illuminate the metaphysics of games more generally. (shrink)

This paper proposes a cognitive linguistic explanation of the unusual narrative construal of time as moving backwards. It shows that backwards time in narrative involves setting up an alternative space in which a second narrative is constructed simultaneously, resulting in a viewpoint hierarchy which postulates four viewpoints on each discourse statement. The paper draws together research on conceptual metaphor, mental spaces theory and viewpoint multiplicity, bringing it to bear on discourse fragments. The majority of these are taken from Martin Amis’s (...) Time’s Arrow (Amis, Martin. 2003 [1991]. Time’s Arrow. London: Vintage.) (a logically consistent and linguistically revealing text), but the discussion is contextualised with further examples. It is argued that the causal construal implied by narrative is not limited by our phenomenological experience, as other studies (e.g., Evans, Vyvyan. 2013. Language and time: A cognitive linguistics approach. Cambridge: Cambridge University Press.) might suggest, but is instead prompted by conceptual re-interpretation of deictic markers in the discourse. The analysis focuses on causal construal, negation (especially causal catachresis), and alternativity. It shows how linguistic features at the sentential level generate top-down reorganization of both episodic structure and discourse meaning, and how this relies on multiplicity of viewpoint and a conceptual “zooming out” prompting the perception of irony. (shrink)

Two justifications of backward induction (BI) in generic perfect information games are formulated using Bonanno's (1992; Theory and Decision 33, 153) belief systems. The first justification concerns the BI strategy profile and is based on selecting a set of rational belief systems from which players have to choose their belief functions. The second justification concerns the BI path of play and is based on a sequential deletion of nodes that are inconsistent with the choice of rational belief (...) functions. (shrink)

Genericity is the idea that the same program can work at many different data types. Longo, Milstead and Soloviev proposed to capture the inability of generic programs to probe the structure of their instances by the following equational principle: if two generic programs, viewed as terms of type , are equal at any given instance A[T], then they are equal at all instances. They proved that this rule is admissible in a certain extension of System F, but finding a semantically (...) motivated model satisfying this principle remained an open problem.In the present paper, we construct a categorical model of polymorphism, based on game semantics, which contains a large collection of generic types. This model builds on two novel constructions:•A direct interpretation of variable types as games, with a natural notion of substitution of games. This allows moves in games A[T] to be decomposed into the generic part from A, and the part pertaining to the instance T. This leads to a simple and natural notion of generic strategy.•A “relative polymorphic product” which expresses quantification over the type variable Xi in the variable type A with respect to a “universe” which is explicitly given as an additional parameter B. We then solve a recursive equation involving this relative product to obtain a universe in a suitably “absolute” sense.Full Completeness for ML types is proved for this model. (shrink)

The recent operationalization of the famous Newcomb's game by Schmidt (1998) offers an interesting and thought-provoking look at the plausibility of backward causation in a Newtonian universe. Hereby we investigate two details of the Schmidt's scenario which may, at least in principle, invalidate his conclusion in two different domains: one dealing with the issue of Newtonian predictability in specific instance of human actions, and the other stemming from a possible strategy aimed at obviating the anthropically oriented view of (...) backward causation as applied to a judicial and ethical problem posed by a version of the scenario. We conclude that the scenario is at least to be more complex than originally presented in order to remain viable. However, it points to a very deep and delicate question of compatibility of backward causation with the conventional ethical standards. (shrink)

We generalize the intuitionistic Hyland–Ong games to a notion of polarized games allowing games with plays starting by proponent moves. The usual constructions on games are adjusted to fit this setting yielding game models for both Intuitionistic Linear Logic and Polarized Linear Logic. We prove a definability result for this polarized model and this gives complete game models for various classical systems: , λμ-calculus, … for both call-by-name and call-by-value evaluations.

The problem of finding sufficient doxastic conditions for backward induction in games of perfect information is analyzed in a syntactic framework with subjunctive conditionals. This allows to describe the structure of the game by a logical formula and consequently to treat beliefs about this structure in the same way as beliefs about rationality. A backward induction and a non-Nash equilibrium result based on higher level belief in rationality and the structure of the game are derived.

We analyze the sequential structure of dynamic games with perfect information. A three-stage account is proposed, that species setup, reasoning and play stages. Accordingly, we define a player as a set of agents corresponding to these three stages. The notion of agent connectedness is introduced into a type-based epistemic model. Agent connectedness measures the extent to which agents' choices are sequentially stable. Thus describing dynamic games allows to more fully understand strategic interaction over time. In particular, we (...) provide suffcient conditions for backward induction in terms of agent connectedness. Also, our framework makes explicit that the epistemic independence assumption involved in backward induction reasoning is stronger than usually presumed, and makes accessible multiple-self interpretations for dynamic games. (shrink)

Rational choice theory enjoys unprecedented popularity and influence in the behavioral and social sciences, but it generates intractable problems when applied to socially interactive decisions. In individual decisions, instrumental rationality is defined in terms of expected utility maximization. This becomes problematic in interactive decisions, when individuals have only partial control over the outcomes, because expected utility maximization is undefined in the absence of assumptions about how the other participants will behave. Game theory therefore incorporates not only rationality but also common (...) knowledge assumptions, enabling players to anticipate their co-players' strategies. Under these assumptions, disparate anomalies emerge. Instrumental rationality, conventionally interpreted, fails to explain intuitively obvious features of human interaction, yields predictions starkly at variance with experimental findings, and breaks down completely in certain cases. In particular, focal point selection in pure coordination games is inexplicable, though it is easily achieved in practice; the intuitively compelling payoff-dominance principle lacks rational justification; rationality in social dilemmas is self-defeating; a key solution concept for cooperative coalition games is frequently inapplicable; and rational choice in certain sequential games generates contradictions. In experiments, human players behave more cooperatively and receive higher payoffs than strict rationality would permit. Orthodox conceptions of rationality are evidently internally deficient and inadequate for explaining human interaction. Psychological game theory, based on nonstandard assumptions, is required to solve these problems, and some suggestions along these lines have already been put forward. Key Words: backward induction; Centipede game; common knowledge; cooperation; epistemic reasoning; game theory; payoff dominance; pure coordination game; rational choice theory; social dilemma. (shrink)

The return of a classic book about games and play that illuminates the relationship between the well-played game and the well-lived life. In The Well-Played Game, games guru Bernard De Koven explores the interaction of play and games, offering players—as well as game designers, educators, and scholars—a guide to how games work. De Koven's classic treatise on how human beings play together, first published in 1978, investigates many issues newly resonant in the era of video and (...) computer games, including social gameplay and player modification. The digital game industry, now moving beyond its emphasis on graphic techniques to focus on player interaction, has much to learn from The Well-Played Game. De Koven explains that when players congratulate each other on a “well-played” game, they are expressing a unique and profound synthesis that combines the concepts of play (with its associations of playfulness and fun) and game (with its associations of rule-following). This, he tells us, yields a larger concept: the experience and expression of excellence. De Koven—affectionately and appreciatively hailed by Eric Zimmerman as “our shaman of play”—explores the experience of a well-played game, how we share it, and how we can experience it again; issues of cheating, fairness, keeping score, changing old games (why not change the rules in pursuit of new ways to play?), and making up new games; playing for keeps; and winning. His book belongs on the bookshelves of players who want to find a game in which they can play well, who are looking for others with whom they can play well, and who have discovered the relationship between the well-played game and the well-lived life. (shrink)

Open peer commentary on the article “Learning about Learning with Teachers and Young Children” by Chrystalla Papademetri-Kachrimani. Upshot: The activities that Papademetri-Kachrimani presents in her stories create situations that lead to unexpected results, thus opening the potential for learning about learning in teachers’ professional development. These integrate modeling-based learning - arguably a form of constructionism -, and allow learners to move back-and-forth between representations in order to develop strategies and rules.

We describe a class of semantic extensive entailment game with algorithmic players, related to game-theoretic semantics, and generalized to classical first-order semantic entailment. Players have preferences for parsimonious spending of computational resources, and compute partial strategies, under qualitative uncertainty about future histories. We prove the existence of local preferences for moves, and strategic fixpoints, that allow to map eeg game-tree to the building rules and closure rules of Smullyan's semantic tableaux. We also exhibit a strategy profile that solves (...) the fixpoint selection problem, and can be mapped to systematic constructions of semantic trees, yielding a completeness result by translation. We conclude on possible generalizations of our games. (shrink)

The mechanisms responsible for the fine tuning of development, where the wildtype phenotype is reproduced with high fidelity, are not well understood. The difficulty in approaching this problem is the identification of mutant phenotypes indicative of a defect in these fine‐tuning control mechanisms. Evolutionary biologists have used asymmetry as a measure of developmental homeostasis. The rationale for this was that, since the same genome controls the development of the left and right sides of a bilaterally symmetrical organism, departures from (...) symmetry can be used to measure genetic or environmental perturbations. This paper examines the relationship between asymmtry and resistance to organophosphorous insecticides in the Australian sheep blowfly, Lucilia cuprina. A resistance gene, Rop‐1, which encodes a carboxylesterase enzyme, also confers a significant increase in asymmetry. Continued exposure of resistant populations to insecticide has selected a dominant suppressor of the asymmetry phenotype. Genetic evidence indicates that the modifier is the L. cuprina Notch homologue. (shrink)

In certain finite extensive games with perfect information, Cristina Bicchieri (1989) derives a logical contradiction from the assumptions that players are rational and that they have common knowledge of the theory of the game. She argues that this may account for play outside the Nash equilibrium. She also claims that no inconsistency arises if the players have the minimal beliefs necessary to perform backward induction. We here show that another contradiction can be derived even with minimal (...) beliefs, so there is no paradox of common knowledge specifically. These inconsistencies do not make play outside Nash equilibrium plausible, but rather indicate that the epistemic specification must incorporate a system for belief revision. Whether rationality is common knowledge is not the issue. (shrink)

This paper explores the cooperation mechanisms for the prisoner’s dilemma game, a canonical example for studying cooperation mechanisms, with Bayesian games. By the approach allowing simultaneous moves with the assumption that the players might be self-interested or norm-following, we establish four possible Bayesian game models, all of which are cooperation mechanisms for the prisoner’s dilemma game except for the model in which one of the two players must be self-interested.

As they are conventionally formulated, Boolean games assume that players make their choices in ignorance of the choices being made by other players – they are games of simultaneous moves. For many settings, this is clearly unrealistic. In this paper, we show how Boolean games can be enriched by dependency graphs which explicitly represent the informational dependencies between variables in a game. More precisely, dependency graphs play two roles. First, when we say that variable x depends (...) on variable y, then we mean that when a strategy assigns a value to variable x, it can be informed by the value that has been assigned to y. Second, and as a consequence of the first property, they capture a richer and more plausible model of concurrency than the simultaneous-action model implicit in conventional Boolean games. Dependency graphs implicitly define a partial ordering of the run-time events in a game: if x is dependent on y, then the assignment of a value to y must precede the assignment of a value to x; if x and y are independent, however, then we can say nothing about the ordering of assignments to these variables—the assignments may occur concurrently. We refer to Boolean games with dependency graphs as partial-order Boolean games. After motivating and presenting the partial-order Boolean games model, we explore its properties. We show that while some problems associated with our new games have the same complexity as in conventional Boolean games, for others the complexity blows up dramatically. We also show that the concurrency in partial-order Boolean games can be modelled using a closure-operator semantics, and conclude by considering the relationship of our model to Independence-Friendly logic. (shrink)

We show that for solvable games, the calculation of the strategies which survive iterative elimination of dominated strategies in normal games is equivalent to the calculation of the backward induction outcome of some extensive game. However, whereas the normal game form does not provide information on how to carry out the elimination, the corresponding extensive game does. As a by-product, we conclude that implementation using a subgame perfect equilibrium of an extensive game with perfect information is (...) equivalent to implementation through a solution concept which we call guided iteratively elimination of dominated strategies which requires a uniform order of elimination. Journal of Economic Literature Classifica-. (shrink)